Finite Group Databasebeta

← C3×#10590522 #13157582

Subgroups — C3×#10590522 #13157582

Order 1728.

678 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64104C211yesmaximal, characteristic
2C3×#63871C211yesmaximal, characteristic, complemented
3C2×C3×#3717C211yesmaximal, characteristic, complemented
4C2×C3×#3712C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6C32×#807C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#4886—31nomaximal
9C3×#4812—31nomaximal
10#10590522C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3752—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×#3722—21no—
22C3×#3717—22no—
23C3×#3712—22no—
24#64104C2×C311yescharacteristic
25C3×#946—31no—
26C2×C3×#215—31no—
27C2×C3×#204—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C3×#897—31no—
31C3×#895—31no—
32C3×#895—31no—
33C3×#895—61no—
34#63871C2×C311yescharacteristic, complemented
35C2×#3717C2×C311yescharacteristic, complemented
36C2×#3712C2×C311yescharacteristic, complemented
37C32×#107—31no—
38C32×#103—31no—
39C3×#821—31no—
40C3×#810—21no—
41C3×#810D311yescharacteristic, complemented
42C3×#810C2×C311yescharacteristic, complemented
43C3×#809—31no—
44C3×#807—21no—
45C3×#807D311yescharacteristic, complemented
46C3×#807C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3D411yescharacteristic
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—42no—
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8D412yescomplemented
64C3×#468—91no—
65#4886—31no—
66#4812—31no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×D12—61no—
76C2×C3×D12—62no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—61no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C2×C32×C8—31no—
84C2×C32×C8—31no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—31no—
89C32×C4⋊C4—41no—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C4⋊C4C22×C311yescharacteristic
93#3752—21no—
94C3×#215—62no—
95C3×(C2×D3)⋊C4—61no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—62no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C22×C311yescharacteristic
104C3×Dic3⋊C4—61no—
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111C3×#204—62no—
112C3×ΓL(1,𝔽25)—62no—
113C3×ΓL(1,𝔽25)—62no—
114Dic3⋊Dic3—21no—
115#3722—21no—
116#3717—22no—
117#3712—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—21no—
129C32×Dic3—41no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×M16—91no—
133C3×C22⋊Q8—91no—
134C3×#121—91no—
135#946—31no—
136C2×#215—31no—
137C2×#204—31no—
138C2×ΓL(1,𝔽25)—31no—
139C2×ΓL(1,𝔽25)—31no—
140#897—31no—
141#895—31no—
142#895—31no—
143#895—61no—
144C3×#107—31no—
145C3×#107—31no—
146C3×#107—61no—
147C3×#103—31no—
148C3×#103—31no—
149C3×#103—61no—
150C3×#102—91no—
151#821—31no—
152#810C3×D311yescharacteristic, complemented
153#809—31no—
154#807C3×D311yescharacteristic, complemented
155C22×C3⋊D3—21no—
156C22×C3⋊D3C3×D411yescharacteristic
157C22×C3×D3—31no—
158C22×C3×D3—31no—
159C22×C3×D3—61no—
160C22×C3×D3—62no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C32×Q8—61no—
164C32×Q8—61no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—42no—
176C2×C32×C4—61no—
177C2×C32×C4—61no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C23×C311yescharacteristic
181C2×C3⋊Dic3C3×D411yescharacteristic
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—42no—
185C4×C3⋊D3—21no—
186C4×C3⋊D3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×D12—121no—
202C3×D12—124no—
203C3×D12—122no—
204C3×D12—122no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—61no—
208C3×C4×D3—61no—
209C3×C4×D3—121no—
210C3×C4×D3—121no—
211C3×Dic6—22no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—41no—
217C32×C8—32no—
218C32×C8—61no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—22no—
222C3×C3⋊C8—41no—
223C3×C3⋊C8D1212yescomplemented
224C3×C3⋊C8C3×D412yescomplemented
225#468—91noSylow
226C2×C33#12111yescharacteristic
227C2×C33AGL(1,ℤ/8)11yescharacteristic
228C2×C33#13711yescharacteristic
229C3×C3⋊D3—41no—
230C3×C3⋊D3—81no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—61no—
234C2×C3×D4—91no—
235C2×C3×D4—181no—
236C22×C3×C4—91no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—61no—
240C2×D12—61no—
241C2×D12—61no—
242C2×D12—62no—
243C2×C4×D3—31no—
244C2×C4×D3—31no—
245C2×C4×D3—61no—
246C2×Dic6C2×C3×D311yescharacteristic
247C3×SD16—182no—
248C3×M16—182no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4—61no—
262C3×C4⋊C4—181no—
263C3×C4⋊C4D3211yescharacteristic
264C3×C4⋊C4C2×C3×D311yescharacteristic
265C3×C4⋊C4C2×C3×D311yescharacteristic
266C3×C22⋊C4—181no—
267C3×C22⋊C4—181no—
268#215—62no—
269(C2×D3)⋊C4—61no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—61no—
272(C2×D3)⋊C4—61no—
273(C2×D3)⋊C4—62no—
274C4⋊Dic3—61no—
275C4⋊Dic3C2×C3×D311yescharacteristic
276Dic3⋊C4—61no—
277C2×C3⋊C8C2×C3×D311yescharacteristic
278C2×C3⋊C8C2×C3×D311yescharacteristic
279#204—62no—
280ΓL(1,𝔽25)—62no—
281ΓL(1,𝔽25)—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32D3×D411yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32C2×C3×D411yescharacteristic
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3×D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—122no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4—42no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—81no—
326C32×C4—81no—
327C32×C4—82no—
328C32×C4—121no—
329C32×C4—121no—
330C32×C4C2×D1211yescharacteristic
331C32×C4#23511yescharacteristic
332C32×C4D3×D411yescharacteristic
333C32×C4#23911yescharacteristic
334C32×C4C2×C3×D411yescharacteristic
335C32×C4C3×#4111yescharacteristic
336C3⋊Dic3—21no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—81no—
345C3×Dic3—121no—
346C2×SD16—91no—
347C2×M16—91no—
348C22⋊Q8—91no—
349#121—91no—
350#107—31no—
351#103—31no—
352#102—91no—
353C33#46811yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C23×C3—181no—
356C22×D3—31no—
357C22×D3—31no—
358C22×D3—61no—
359C22×D3—62no—
360C22×D3—61no—
361C22×D3—61no—
362C3×Q8—62no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—61no—
366C3×Q8—61no—
367C3×Q8—121no—
368C3×D4—181no—
369C3×D4—181no—
370C3×D4—362no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—41no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—91no—
387C2×C3×C4—121no—
388C2×C3×C4—121no—
389C2×C3×C4—181no—
390C2×C3×C4—181no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C22×C3×D311yescharacteristic
393C2×C3×C4C22×C3×D311yescharacteristic
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—61no—
399D12—61no—
400D12—61no—
401D12—61no—
402D12—61no—
403D12—121no—
404D12—121no—
405D12—124no—
406D12—122no—
407D12—122no—
408C4×D3—61no—
409C4×D3—61no—
410C4×D3—61no—
411C4×D3—61no—
412C4×D3—121no—
413C4×D3—121no—
414Dic6—21no—
415Dic6—21no—
416C3×C8—32no—
417C3×C8—32no—
418C3×C8—62no—
419C3×C8—61no—
420C3×C8—61no—
421C3×C8—121no—
422C3⋊C8—21no—
423C3⋊C8C3×D1212yescomplemented
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32#88411yescharacteristic
437C2×C32#89511yescharacteristic
438C2×C32#91111yescharacteristic
439C2×C32#91411yescharacteristic
440C2×C32#91811yescharacteristic
441C2×C32#91811yescharacteristic
442C2×C32C3×#12111yescharacteristic
443C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
444C2×C32C3×#13711yescharacteristic
445C3⋊D3—41no—
446C3⋊D3—81no—
447C3×D3—122no—
448C3×D3—121no—
449C3×D3—121no—
450C3×D3—242no—
451C3×D3—241no—
452C3×D3—241no—
453C2×Q8—31no—
454C2×D4—91no—
455C2×D4—181no—
456C22×C4—91no—
457SD16—182no—
458M16—182no—
459C2×C8—31no—
460C2×C8—31no—
461C4⋊C4—31no—
462C4⋊C4—181no—
463C4⋊C4C3×D3211yescharacteristic
464C22⋊C4—181no—
465C22⋊C4—181no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—181no—
471C22×C3—181no—
472C22×C3—181no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3#381111yescharacteristic
477C22×C3C3×D3×D411yescharacteristic
478C22×C3C3×D3×D411yescharacteristic
479C2×D3—62no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—82no—
512C3×C4—122no—
513C3×C4—121no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—121no—
517C3×C4—181no—
518C3×C4—241no—
519C3×C4#379911yescharacteristic
520C3×C4D3×D1211yescharacteristic
521C3×C4C2×C3×D1211yescharacteristic
522C3×C4C3×#23511yescharacteristic
523C3×C4C3×D3×D411yescharacteristic
524C3×C4C3×#23911yescharacteristic
525Dic3—41no—
526Dic3—41no—
527Dic3—61no—
528Dic3—61no—
529Dic3—121no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#481211yescharacteristic, complemented
535C32#488611yescharacteristic, complemented
536C32C3×#46811yescharacteristic, complemented
537C23—91no—
538C23—181no—
539Q8—61no—
540Q8—61no—
541D4—181no—
542D4—181no—
543D4—362no—
544C2×C4—21no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—91no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4C2×C3×D3211yescharacteristic
551C8—32no—
552C8—61no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—361no—
566C2×C3—721no—
567C2×C3#6391011yescharacteristic
568C2×C3#6393711yescharacteristic
569C2×C3#6394511yescharacteristic
570C2×C3C3×#88411yescharacteristic
571C2×C3C3×#89511yescharacteristic
572C2×C3C3×#91111yescharacteristic
573C2×C3C3×#91411yescharacteristic
574C2×C3C3×#91811yescharacteristic
575C2×C3C3×#91811yescharacteristic
576D3—122no—
577D3—121no—
578D3—121no—
579D3—242no—
580D3—241no—
581D3—241no—
582C22—181no—
583C22—181no—
584C22—181no—
585C22—361no—
586C22—361no—
587C22—361no—
588C22C3×#381111yescharacteristic
589C4—41no—
590C4—121no—
591C4—121no—
592C4—181no—
593C4C3×#379911yescharacteristic
594C4C3×D3×D1211yescharacteristic
595C3—21no—
596C3—21no—
597C3—21no—
598C3—41no—
599C3#1059052211yescharacteristic, direct product factor
600C3C3×#481211yescharacteristic, complemented
601C3C3×#488611yescharacteristic, complemented
602C2—361no—
603C2—721no—
604C2C3×#6391011yescharacteristic
605C2C3×#6393711yescharacteristic
606C2C3×#6394511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#63871C2semidirect
3. C2×C3×#3717C2semidirect
4. C2×C3×#3712C2semidirect
5. C32×#810C2semidirect
6. C32×#807C2semidirect
7. C3×#63564C2semidirect
10. #10590522C3direct (N × H)
34. #63871C2×C3semidirect
35. C2×#3717C2×C3semidirect
36. C2×#3712C2×C3semidirect
41. C3×#810D3semidirect
42. C3×#810C2×C3semidirect
45. C3×#807D3semidirect
46. C3×#807C2×C3semidirect
47. #63564C2×C3semidirect
63. C32×C3⋊C8D4semidirect
152. #810C3×D3semidirect
154. #807C3×D3semidirect
223. C3×C3⋊C8D12semidirect
224. C3×C3⋊C8C3×D4semidirect
353. C33#468semidirect
423. C3⋊C8C3×D12semidirect
534. C32#4812semidirect
535. C32#4886semidirect
536. C32C3×#468semidirect
599. C3#10590522direct (N × H)
600. C3C3×#4812semidirect
601. C3C3×#4886semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#641042
C3×#638712
C2×C3×#37172
C2×C3×#37122
C32×#8102
C32×#8072
C3×#635642
C3×#48863
C3×#48123
#105905223